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President Obama’s failed Olympic gambit Friday was a blow to his image on the world stage and a very public humbling experience for a man who has grown unaccustomed to losing elections.

After carving out 20 hours to try and secure the games for his hometown of Chicago - about 15 of them spent in the air between Washington and Copenhagen, where the International Olympic Committee ultimately voted for Rio de Janeiro to host the 2016 Summer Olympics - Mr. Obama emerged sheepishly from Marine One Friday afternoon and loped, head down, across the South Lawn. During brief remarks a few moments later, he congratulated the people of Brazil, and expressed no regrets.

“I believe it’s always a worthwhile endeavor to promote and boost the United States of America,” he said.

Whether the Copenhagen defeat will carry lasting implications for Mr. Obama’s image back home, though, remained a subject of intense debate even after the president returned to the Oval Office with a commitment to focus on more substantive matters, such as the latest flagging employment numbers.

To many of the president’s allies, the episode represented a brief political diversion that was worthwhile on the chance it could elevate Chicago’s bid at a time when Brazil was emerging as a sentimental favorite.

To Mr. Obama’s critics, the trip resonated as a sign that the White House has begun to overestimate the value of Mr. Obama’s personal magnetism and the power of a well-delivered speech. The episode was, in the view of Republican strategist Todd Harris, something the president might call a teachable moment.

“I think there’s an arrogance that permeates just about everything the Obama administration does,” Mr. Harris said. “I don’t think the IOC’s decision will hurt him politically, but it is illuminating and hopefully will be a learning experience about how much influence he does or does not have abroad.”

Inside the White House, the decision to jet to Copenhagen to make a last-ditch appeal to the members of the International Olympic Committee, while not the original plan, began to look like a worthwhile risk. From the vantage point of seasoned Olympics watchers, even Republicans like former Massachusetts Gov. Mitt Romney, the move seemed inspired.

“We saw a plane, President Obama’s plane, Air Force One, arriving and the television was showing all the time President Obama. And my friends were saying ‘Oh, we’ve lost now. President Obama has arrived and we’re going to lose,’” he said Friday in an emotional interview.

But the high expectations only made it look worse when Chicago’s bid was knocked out in the first round of voting.

“The president put his prestige on the line and to have him lose, it was really a slap in the face to the United States,” said John Feehery, who served as top aide to former Republican House Speaker J. Dennis Hastert. “I do think this is instructive about how seriously the international community takes Obama. It seems they don’t take him very seriously.”

Mr. Obama’s supporters said the sting of the defeat will only be momentary.

Robert Zimmerman, a public relations executive and Democratic National Committee member from New York, said the 24/7 news cycle will initially draw attention to the defeat.

“This will be a major story for the pundits, and it will be replaced in the next 24 hours by something else,” Mr. Zimmerman said. “That’s the strength and weakness of the 24/7 news cycle. Nothing is forever, neither the good news nor the disappointments.”

Democratic strategist Howard Wolfson said the outcome reminded him of the trip New York Mayor Michael R. Bloomberg made four years ago in an attempt to secure an Olympic bid for his city.

Mr. Wolfson, now an adviser to Mr. Bloomberg, said when the mayor returned empty-handed, voters still thanked him for the effort.

“I think it was right for President Obama to go and make the pitch,” Mr. Wolfson said. “They clearly went in another direction. But the president needs to be able to go and make a strong case for the country.”

How the public responds may help future presidents assess whether the Olympic bid process is an appropriate stage for an American president, said Dana Perino, who served as press secretary to former President George W. Bush.

Unlike other countries, Ms. Perino said, “the tradition in America has been that we don’t use our government to influence our Olympic team, and presidents typically leave it to the private sector to win that bid.”

But she, for one, does not fault Mr. Obama for breaking with that tradition and making the effort.

“I think President Obama sincerely wanted to help his hometown and the country to get the Olympics because it would be the best place for them to be held,” she said. “He decided to take the risk. And unfortunately for Chicago, it didn’t work.”